Smalltalk-80: the language and its implementation
Smalltalk-80: the language and its implementation
On understanding types, data abstraction, and polymorphism
ACM Computing Surveys (CSUR) - The MIT Press scientific computation series
Polymorphic type inference and containment
Information and Computation - Semantics of Data Types
An extension of system F with subtyping
Information and Computation - Special conference issue: international conference on theoretical aspects of computer software
A Sequent Calculus for Subtyping Polymorphic Types
MFCS '96 Proceedings of the 21st International Symposium on Mathematical Foundations of Computer Science
ECOOP '95 Proceedings of the 9th European Conference on Object-Oriented Programming
Equational Axiomatization of Bicoercibility for Polymorphic Types
Proceedings of the 15th Conference on Foundations of Software Technology and Theoretical Computer Science
The subtyping problem for second-order types is undecidable
Information and Computation
LICS '95 Proceedings of the 10th Annual IEEE Symposium on Logic in Computer Science
Type systems for object-oriented programming languages
Type systems for object-oriented programming languages
A Subtyping for Extensible, Incomplete Objects
Fundamenta Informaticae
On the building of affine retractions
Mathematical Structures in Computer Science
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This paper is devoted to a comprehensive study of polymorphic subtypes with products. We first present a sound and complete Hilbert style axiomatization of the relation of being a subtype in presence of →, x type constructors and the ∀ quantifier, and we show that such axiomatization is not encodable in the system with →, ∀ only. In order to give a logical semantics to such a subtyping relation, we propose a new form of a sequent which plays a key role in a natural deduction and a Gentzen style calculi. Interestingly enough, the sequent must have the form E ⊢ T, where E is a non-commutative, non-empty sequence of typing assumptions and T is a finite binary tree of typing judgements, each of them behaving like a pushdown store. We study basic metamathematical properties of the two logical systems, such as subject reduction and cut elimination. Some decidability/undecidability issues related to the presented subtyping relation are also explored: as expected, the subtyping over →, x, ∀ is undecidable, being already undecidable for the →, ∀ fragment (as proved in [15]), but for the x, ∀ fragment it turns out to be decidable.